Multiplicative bijections of semigroups of interval-valued continuous functions

We characterize all compact and Hausdorff spaces $X$ which satisfy the condition that for every multiplicative bijection $\varphi$ on $C(X, I)$, there exist a homeomorphism $\mu : X \longrightarrow X$ and a continuous map $p: X \longrightarrow (0, +\infty)$ such that

$$\varphi (f) (x) = f(\mu (x))^{p(x)}$$

for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. 134 (2006), 1065-1075). Some related results on other semigroups of functions are also given.